## Without Care, Average Rates of Return Don’t Mean Anything

Total return depends on several factors.  The rate of earning, the capital employed and the time it is employed.  Thus 5% per annum means for each \$100 of capital the return is \$5.00 per year.  Pretty obvious.  The problems come when I start treating the 5% number as an absolute.

It is not an absolute it is a time related ratio.

You must notice that you cannot average ratios or rates of return.  Perhaps an example from something other than finance will help.

You are driving along in an old, poorly maintained car.  It wish to maintain an average speed of 60 mph.  The car can travel that speed on the flat ground but only 30 mph going up a hill.  In your path is a hill which is exactly one mile from bottom to top and another mile from top to bottom.  Having gone up at 30 mph, how fast would the car need to travel down the hill to average 60 mph for the two-mile hill?

Hint:  90 mph is wrong.

Sure the average of 90 and 30 is 60 but, that does not matter.  You cannot average rates of change.

The correct answer is that it cannot be done.  At 30 mph it takes 2 minutes to travel one mile to the top of the hill.  To average 60 mph, you must cover 2 miles in 2 minutes.  Therefore, there is no speed that will get you to the bottom soon enough.  Your two minutes has been used up to reach the top.

If you average stock market returns, you will be mislead.  Lose 10% each year for 2 years and then make 10% for the next two, does not average out to zero rate of return.  It results in a loss of about 1% annually.

If I invest \$1,000 and get \$1,500 back in 10 years, I have not earned 5%.   4.14% instead.

To get average rates of return that means something, you will be forced to do arithmetic.  In the 10 year example, I get back 1.5 times my capital in 10 years so my rate of return is the tenth root of 1.5 or 4.14%.  There are calculators,  readily available as a phone app, that will handle that calculation.  Or Excel.

People tend to make the mistake of averaging the rate of return without considering that compound interest works differently than simple interest and compound interest is the one that matters.  You think about it by using the time factor to control.  How much do I get back over how much did I put in.  The take the nth root.

It surprises most people how wrong intuition is over long periods.

If someone bought a cottage on Clear Lake in 1956 for \$2,500 they might be amazed to find that the \$750,000 it is worth now is a rate of return less than 10%.  My parent should have purchased many, but the 10% doesn’t seem thrilling until you have held the asset for a long time.  By 1976, half way, it was worth just \$43,000.

Compound interest is not intuitive.  Learn to do the arithmetic and get tools that will allow you to do so.  At a minimum become familiar with the rule of 72.

Don Shaughnessy arranges life insurance for people who understand the value of a life insured estate. He can be reached at The Protectors Group, a large insurance, employee benefits, and investment agency in Peterborough, Ontario.  In previous careers, he has been a partner in a large international public accounting firm, CEO of a software start-up, a partner in an energy management system importer, and briefly in the restaurant business.